Find the indefinite integral as indicated.
Integral from nothing to nothing StartFraction d Over dx EndFraction left parenthesis x Superscript 8 Baseline plus 4 x squared plus 4 right parenthesis dx
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Part 1
Integral from nothing to nothing StartFraction d Over dx EndFraction left parenthesis x Superscript 8 Baseline plus 4 x squared plus 4 right parenthesis dx
equals

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To find the indefinite integral of \(\frac{d}{dx} (x^8 + 4x^2 + 4)\) with respect to \(x\), we first simplify the expression inside the integral.

### Step 1: Differentiate the Inner Function
Differentiate \(x^8 + 4x^2 + 4\) with respect to \(x\):

\[
\frac{d}{dx}(x^8 + 4x^2 + 4) = 8x^7 + 8x
\]

### Step 2: Integrate the Result
Now, we integrate the resulting expression \(8x^7 + 8x\) with respect to \(x\):

\[
\int (8x^7 + 8x) \, dx
\]

### Step 3: Apply the Power Rule for Integration
Using the power rule for integration, we integrate each term separately:

1. \(\int 8x^7 \, dx = 8 \cdot \frac{x^8}{8} = x^8\)
2. \(\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2\)

Combining these results:

\[
x^8 + 4x^2 + C
\]

where \(C\) is the constant of integration.

### Final Result
The indefinite integral is:

\[
\int \frac{d}{dx} \left( x^8 + 4x^2 + 4 \right) \, dx = x^8 + 4x^2 + C
\]

This means that integrating the derivative of a function essentially gives us back the original function, up to an added constant of integration. If you have any further questions or need additional details, feel free to ask! 

Here are some related questions to explore this topic further:

1. What is the significance of the constant of integration?
2. How do we differentiate and integrate polynomial functions?
3. What are the applications of indefinite integrals in physics?
4. How does the Fundamental Theorem of Calculus relate to this problem?
5. What are some common techniques for integrating more complex functions?
6. How do definite and indefinite integrals differ?
7. Can all functions be integrated indefinitely?
8. How does integration relate to finding the area under a curve?

**Tip:** When differentiating and integrating functions, always double-check your work by differentiating your integral to ensure it matches the original function.

Learn how to calculate the indefinite integral of x^8 + 4x^2 + 4 step-by-step. This problem involves applying the power rule for integration and understanding the Fundamental Theorem of Calculus. Suitable for undergraduate students and anyone interested in calculus.

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